If you consider low-cost airlines, those usually price each segment individually based on demand for that specific flight, so whether you include a Saturday evening or not should not change the total price.
On traditional airlines, there are usually different fares for one-way and return trips (and multi-city, open jaws, etc.). Their traditional model is to try to make business travellers pay more (the company pays, and it supposedly has deep pockets), and leisure travellers pay less (they have less money to spend).
So they have lots of rules to try to match this distinction: business travellers need flexibility (exchange/refund tickets, get one-way tickets, "open" tickets, book at the last minute, don't want to stay over the week-end, often make very short trips...) while leisure travellers are more price-conscious and don't mind booking ages in advance, stay over the week-end, or a long time, etc.
They then create lots of different fares, each with its own set of rules, and a different price. The fare is then associated to a "bucket" which determines availability on each flight.
So, yes, there are cheap(er) fares which require you to fulfil one or more of these conditions:
- book a round-trip
- stay the night from Saturday to Sunday
- stay at least 7 nights, or 14 nights
- book at least 3, 7, 14, 21, 30, 45 or 60 days in advance
- not be eligible for refunds or exchanges, or have to suffer penalties for it
- and probably a few more
Note that I'm not aware of rules asking to stay two Saturday-to-Sunday nights (though there may be), but there are rules with a minimum stay duration.
Fares including advance purchase, minimum stay and/or Saturday-to-Sunday night, no refunds or exchanges, or penalties for doing so, are (or were?) often called "APEX fares" (Advance Purchase Excursion).
Some advanced tools (such as www.expertflyer.com) allow you to list fares for a given city pair. You'll have the price, the fare rules (those can be quite lengthy and hard to read for the uninitiated), and the fare bucket. Then you can try to find flights with availability in the requested fare bucket, combining them correctly so that you match the associated fare rules.
Welcome to the world of revenue management!